In Linear Algebra, the matrix-vector product $Ax = B$ serves as a formal shorthand representing a system where vector $\mathbf{b} \in \mathbb{R}^m$ is defined as a linear combination of the columns of an $M \times N$ coefficient matrix $\mathbf{A}$ weighted by components from a vector $\mathbf{x} \in \mathbb{R}^N$. This mechanism encapsulates the theory that matrices function as operators mapping vectors from an $n$-dimensional domain space to an $m$-dimensional codomain, effectively modeling linear transformations. The abstraction eliminates verbose summation notation in favor of compact symbolic representation while preserving the fundamental rules of scalar multiplication and vector addition inherent to vector spaces.
In Linear Algebra, the matrix-vector product $Ax = B$ serves as a formal shorthand representing a system where vector $\mathbf{b} \in \mathbb{R}^m$ is defined as a linear combination of the columns o…